Vortex line nucleation of first-order phase transitions in early universe

Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics

Volumn 460, Issue 1-2, 1999, Pages 36-40

  Kleinert, H ^a  

^a Institut fuer Theoretische Physik der   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0041865215     PISSN: 03702693     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0370-2693(99)00772-8     Document Type: Article

References (36)

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3. For reviews see S. Coleman, Phys. Rev. D 15 (1977) 2929; A. Zichichi (Ed.), also in The Whys of Subnuclear Physics, Ence Lectures 1977, Plenum Press, 1979; I. Afflek, Phys. Rev. Lett. 46 (1981) 391; H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, 2nd ed., World Scientific, Singapore, 1995 (www.physik.fuberlin.de/kleinert/b3).
4. For reviews see S. Coleman, Phys. Rev. D 15 (1977) 2929; A. Zichichi (Ed.), also in The Whys of Subnuclear Physics, Ence Lectures 1977, Plenum Press, 1979; I. Afflek, Phys. Rev. Lett. 46 (1981) 391; H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, 2nd ed., World Scientific, Singapore, 1995 (www.physik.fuberlin.de/kleinert/b3).
5. For reviews see S. Coleman, Phys. Rev. D 15 (1977) 2929; A. Zichichi (Ed.), also in The Whys of Subnuclear Physics, Ence Lectures 1977, Plenum Press, 1979; I. Afflek, Phys. Rev. Lett. 46 (1981) 391; H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, 2nd ed., World Scientific, Singapore, 1995 (www.physik.fuberlin.de/kleinert/b3).
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13. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
14. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
15. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
16. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
17. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
18. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
19. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
20. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
21. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
22. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
23. G. 't Hooft, Nucl. Phys. B 190 (1981) 455; see also: S. Maedan, T. Suzuki, Prog. Theor. Phys. 81 (1989) 229; M. Baker, J.S. Ball, F. Zachanasen, Phys. Rep. 209 (1991) 73; M. Mathur, H.S. Sharatchandra, Phys. Rev. Lett. 66 (1991) 3097; K. Lee, Phys. Rev. D 48 (1993) 2493; M. Sato, S. Yahikozawa, Nucl. Phys. B 436 (1995) 100; E.T. Akhmedov, M.N. Chernodub, M.I. Polikarpov, M.A. Zubkov, Phys. Rev. D 53 (1996) 2087; M.N. Chernodub, M.I. Polikarpov, preprint, 1997, hep-th/9710205. A. Di Giacomo, M. Mathur, Nucl. Phys. B 531 (1998) 302, hep-th/9806153; D. Antonov, D. Ebert, Phys. Lett. B 444 (1998) 208, hep-th/9809018; D.A. Komarov, M.N. Chernodub, JETP Lett. 68 (1998) 117; K.-I. Kondo, Phys. Rev. D 57 (1998) 7467.
24. M.G. Alford, K. Benson, S. Coleman, J. March-Russell, Phys. Rev. Lett. 64 (1990) 1632;
25. 65 (1990) 668 (E); Nucl. Phys. B 349 (1991) 414.
26. 65 (1990) 668 (E); Nucl. Phys. B 349 (1991) 414.
27. See Chapter 13 in the textbook [12] (www.physik.fuberlin.de/kleinert/b1/gifs/v1-716.html).
28. W. Janke, H. Kleinert, Phys. Rev. Lett. 57 (1986) 279 (www.physik.fu-berlin.de/kleinert/130).
29. W. Janke, H. Kleinert, Nucl. Phys. B 270 [FS16] (1986) 399 (www.physik.fu-berlin.de/kleinert/139).
30. H. Kleinert, Gauge Fields in Condensed Matter, vol. I, pp. 531-535 (www.physik.fu-berlin.de/kleinert/b1/gifs/v1-531.html).
31. n are found on p.394 of Ref. [12] (www.physik.fu-berlin.de/kleinert/b1/gifs/v1-394.html). For 14 < n < 22 the numbers have been calculated by P. Butera, M. Comi, (to be published). I am grateful to the authors for communicating these numbers to me prior to publication.
32. W. Janke, H. Klemert, Nucl. Phys. B 270 (1986) 135 (www.physik.fu-berlin.de/kleinert/126)
33. See Chapter 5 in the textbook [12] (www.physik.fuberlin.de/kleinert/b1/gifs/v1-409.html).
34. See p. 530 in the textbook [12] (www.physik.fuberlin.de/kleinert/b1/gifs/v1-530.html).
35. See Chapter 17, in: H. Kleinert, Path Integrals in Quantum Mechanics, Statistics and Polymer Physics, World Scientific, Singapore, 1995, 2nd extended edition (www.physik.fuberlin.de/kleinert/b3).
36. N.D. Atunes, L.M.A. Bettencourt, W.H. Zurek, Phys. Rev. Lett. 82 (1999) 2824-2827.